"Find x_0 and the largest interval, I, for which y(x) is a sol[...]" 1. The problem statement, all variables and given/known data Given that y = –2/x + x is a solution of the differential equation xy' + y = 2x, find ##x_0## and the largest interval, I, for which y(x) is a solution of the initial-value problem: xy' + y = 2x; y(##x_0##) = 1 2. Relevant equations y(x) = –2/x + x y(##x_0##) = 1 3. The attempt at a solution The largest intervals for which y(x) and y'(x) are analytic are (–∞, 0) and (0, ∞), but what is/are the largest interval for which y(x) and y'(x) are analytic for which y(x) is also a solution to the given initial–value problem? Basically, how does forcing y(x) to be a solution to the given initial–value problem affect the interval? Also, if one has two infinite intervals, aren't they equal (rather than one being larger)? For example, isn't (–∞, 0) just as large as (–∞, –25)? (I arbitrarily chose the number –25.) Any help in understanding how to answer this problem correctly would be GREATLY appreciated!